Existence of multiple solutions for second-order discrete boundary value problems
The authors provide sufficient conditions for the existence of (at least) three positive solutions of the discrete two-point boundary value problem \[ \Delta^2 y_{k-1} + f(k,y_k,\Delta y_{k-1})=0, \quad k=1,2,\dots,n, \] \[ y_0=0=y_n, \] where \(\Delta y_k:=y_{k+1}-y_k\) is the usual forward difference operator (note a slightly modified notation used by the authors) and where \(f:\{1,2,\dots,n\}\times{\mathbb R}^2\to{\mathbb R}\) is continuous. The proof is based on the existence of pairs of discrete lower solutions \(\alpha_1\), \(\alpha_2\) and discrete upper solutions \(\beta_1\), \(\beta_2\) that satisfy the inequalities \(\alpha_1\leq\alpha_2\) and \(\beta_1\leq\beta_2\). This method is a discrete analog of the authors' continuous-time results [J. Differ Equations 166, No. 2, 443-454 (2000; Zbl 1013.34017)]. However, in this paper the assumptions do not require \(\alpha_2\leq\beta_1\) and allow \(f\) being dependent on \(\Delta y_{k-1}\). If \(f\) is a function of its second variable \(y_k\) only, then the results of this paper are sharp and generalize those of \textit{R. J. Avery} and \textit{A. C. Peterson} [Panam. Math. J. 8, No. 3, 1-12 (1998; Zbl 0959.39006)]. This paper will be useful for researchers interested in two-point boundary value problems and/or their positive solutions.
discrete two-point boundary value problem
scientific article; zbMATH DE number 1781473